The invention relates to solid-body stress analysis and, more specifically, to the analysis of uniform stress due to compression, stretching, shearing and/or pure bending individually or in combination, of a two-dimensional solid of uniform thickness.
Stress analysis is used primarily in the design of solid structures such as buildings, bridges, dams, and automobiles, ships and aircraft, for determining structural behavior and for ascertaining structural viability and integrity under anticipated or foreseeable loads. Another important field of application is in bioengineering, for hard and soft tissue.
The analysis may involve the use of an abstract or mathematical "model" for the representation of the structural system and loads. In classical analytical idealization, partial differential equations are used. For example, stress in a dam under a hydrostatic load can be described by an elliptic partial differential equation in two spatial dimensions.
As boundary geometry of structural systems is usually complicated, the partial differential equations of structural mechanics typically cannot be solved in the closed analytical exact form. Numerical approximations are sought instead. In one approach, derivatives are replaced with finite differences. Other methods are based on finding an approximation as a linear combination of preassigned functions such as polynomials or trigonometric functions. Also, after a domain or a boundary of interest has been subdivided in the manner of a tiling or tessellation, a piece-wise approximation can be sought according to the finite element method or the boundary element method, respectively. For example, in the case of a two-dimensional domain or surface, triangles can be used for complete tessellation. More generally, "element" refers to any suitable sub-domain of a domain or a boundary portion of interest.
While triangulation is effective and well-understood for compression, shearing and stretching, it tends to incur unacceptably large approximation error for bending. Instead, if quadrilaterals could be used throughout the domain or in a sub-domain, accuracy would be improved. While using a quadrilateral as a straightforward replacement of two triangles, it has been found that an additional approximation error arises due to artificial "stiffening", or additional zero-energy modes are encountered other than rigid-body responses. In the interest of computational accuracy and efficient design, new finite elements are needed which are free of such modeling errors. Desired elements have been termed "defect-free".
The following publications are listed for further background and for an introduction to the computer programming language "Mathematica":
R. H. MacNeal, "A Theorem Regarding the Locking of Tapered Four-noded Membrane Elements", International Journal of Numerical Methods in Engineering, Vol. 24, pp. 1793-1799, 1978;
O. C. Zienkiewicz et al., The Finite Element Method (4th edition), Vols. 1 and 2, McGraw-Hill, New York, 1989;
R. H. MacNeal, Finite Elements: Their Design and Performance, Marcel Dekker, New York, 1994;
S. Wolfram, The Mathematica Book, (3rd edition), Cambridge University Press, New York, 1996.